3.17 Cauchy random variable We recall from Section 3.3.3 that random variable X1 has a Cauchy pdf if

pg170b.gif (2653 bytes)

a. Suppose that S2 X1 + X2, where X1 and X2 are independent Cauchy random variables, each having pdf fxi(.). Using the integral identity

pg170c.gif (5184 bytes)

show that S2 has a pdf 2/[Pi.gif (60
bytes)(4 + y2)].

b. Proceeding by induction, show that

Sn ident.gif (52 bytes) X1 + X2 + . . . + Xn. (all Xi independent)

has a pdf n / [Pi.gif (60 bytes)(n2 + Y2)].

c. Thus, verify that the average of n independent Cauchy samples (i.e., Vn ident.gif (52 bytes) Sn / n) has a Cauchy pdf 1 / [Pi.gif (60 bytes)(n2 + Y2 )]. Thus, "averaging together" a number of independent Cauchy samples yields a pdf for the average identical to that of any one of the individual samples. (This result contrasts sharply to most random variables, for which averaging of n independent samples reduces the variance by a factor of n^-1.)